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%\journal{Computers and Mathematics with Applications}
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\title{Experiment and the Lattice-Boltzmann method study of multi-phase flow in micro-models}
\author[jy]{Jianhui Yang}
\ead{jianhui.yang10@imperial.ac.uk}
\author[jy]{Emily Chapman}
\ead{emily.chapman06@imperial.ac.uk}
\author[jy]{Edo Boek\corref{cor1}}%\fnref{fn1}
\ead{e.boek@imperial.ac.uk}

\cortext[cor1]{Corresponding author}
%\fntext[fn1]{This is the specimen author footnote.}
\address[jy]{Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, United Kingdom}


%\author{Dagwood Remifa\thanks{Thanks to the editors of this wonderful journal!}\\
%\small Department of Inconsequential Studies\\[-0.8ex]
%\small Solatido College, North Kentucky, USA\\
%\small \texttt{remifa@dis.solatido.edu}\\
%\and
%Forgotten Second Author\\
%\small School of Hard Knocks\\[-0.8ex]
%\small University of Western Nowhere\\[-0.8ex]
%\small Nowhere Uvherdov\\
%\small \texttt{no1remembers@me.woe.edu}
%}

%\date{\dateline{Jan 1, 2009}{Jan 2, 2009}{Jan 3, 2009}\\



\begin{abstract}
        In this paper, the spontaneous imbibition of a decane/air system in two sets of micro-models with increasing complexity: a single junction with a squared pore and a pore structure based on actual Berea sandstone were studied using micro fluid experiments and optimised lattice Boltzmann method simulations. The simulations were successfully validated against the micro-fluid experiments on the displacement process of imbibitions. The results confirm our hypothesise: the current capillary pressure filling rules used by the network modelling may not valid for the case of imbibition, instead the local geometry of the network model junction may play an important role. The excellent agreements of interfaces displacement illustrate that the LBM simulation is quite satisfying and adequate for the purposes of simulating multi-component fluid systems coupled with complex geometry, wettability and interfacial tensions.

\end{abstract}


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\begin{keyword}
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the lattice Boltzmann method, spontabeous imbibition, capillary filling rules, micro-models 

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%% or \MSC[2008] code \sep code (2000 is the default)

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\end{frontmatter}

\section{Introduction}

Concern about global warming and emission of anthropogenic $CO_2$ have become crucial issue nowadays, and as result much research in recent years has focused on the implementation of carbon capture and sequestration where $CO_2$ is captured and stored in oil and gas reservoirs. To better understand the physical processes involved in sequestration, we need to understand the pathways selected by the injected $CO_2$ in the reservoirs. The prediction of migration pathways the $CO_2$ will follow is crucial, however, theses predictions are very challenging to attain due to various factors such as: different geological and physical characteristics at each injection site and wetttability alterations the rocks can undergo due to exposure to supercritical $CO_2$ over time (\cite{chalbaud_2007}). 

The prediction of the migration pathway can be investigated by experiments with micro-models or computer simulations such as network modeling and the lattice Boltzmann method. One of the most fundamental research on the fluid displacement in pore-scale using mico-models was carried out by Lenormand et al (\cite{lenormand_1983}). A small network comprising of simple rectangular cross-sectional ducts was used, the channel with the highest capillary pressure was found filled first.  This work has been used as the basis of the network modelling simulation. Chang et al (\cite{chang_2009}) conducted research into snap-off displacement mechanisms by conducting experiments in acrylic micro-models. The imbibitions were found showed good correlation to Lenormand and Zarcone's formula (\cite{lenormand_1983}) for a system where a large number of saturated pores and throats were positioned nearby to a throat fulfilling the necessary threshold pressure. However, this was not the case when there are few saturated pores or throats close to a throat. In addition to these fundamental research, some researchers use network micro-models to study the oil recovery by water and surfactant injection. Sohrabi et al (\cite{sohrabi_2000,sohrabi_2011}) have utilised a glass etched network to investigate oil recovery by water injection.They observed an additional oil recovery of 16\% due to the carbonated water injection. Jamaloei et al (\cite{jamaloei_2010}) studied displacement mechanisms of dilute surfactant flooding in oil-wet and water-wet porous media, via etched glass micro-models. They found that after water-flooding and surfactant flooding the final oil recovery value was greater in the water-wet system than the oil-wet model. Although micro-network model experiments can provide an excellent way of visualising fluid displacement and interaction phenomena at pore scale, more research needs to be conducted to improve the understanding of the basic physical parameter. This can be achieved through investigations on the effect of pore shape on fluid displacement, which is a key parameter in network models, has yet to be exhaustively resolved. However, to the author's knowledge, little attention has been paid on this topic from both experimental and numerical aspect. 

In this paper, we studied the fluid displacement in a single junction micro-model with a squared pore and apart from that, we also investigate a design based on actual rock section: Berea sandstone. This aimed to find the comprehensive solution to the question of whether the pressure filling rules which was used by network modelling capture the fluid processes occurring in actual rock formations. Experiments and the lattice Boltzmann simulations were performed to study the spontaneous imbibitions of a decane/air system in these micro-models. We found that, for the case of imbibitiion, the network modelling filling rules may not valid. The local geometry of the network model junction play an important role. Moreover, the experimental data was compared to computer simulations. The simulations results agrees very well with experimental data that confirmed the ability of the lattice Boltzmann method on complex flow simulation in pore scales. 

\section{Methodology}
Two micro-models of 2.5 dimensionality (they have a finite depth of $75\mu m$) were designed to study the fluid flow in porous media, with increasing complexity ranging from a simple squared pore to a pore structure representing actual Berea sandstone. The pattern with typical scales were showed in Figure (\ref{initial_config}). The oil-wetting micro-models were fabricated in PMMA by Epigem Ltd, a Zeiss inverted microscope (AXIP Observer A1.M) was used to capture still images and the displacement process was captured by a high speed video microscope (FastCam MC2.1, Photron) that has a maximum frame rate of 10,000 frames per second (fps) and a resolution of 512x512 up to a recording setting of 2,000fps. 
\begin{figure}[H]
\begin{center}
\includegraphics[width=3in]{initial_configaration_pics2.eps}
\end{center}
\caption{Single Junction Designs}\label{initial_config}
\end{figure}


The lattice Boltzmann method \cite{doolen_1990,chen_1992,dhumieres1992,lallemand2000} was used to calculate the flow in micro-models. This novel CFD method is able to handle extremely complex geometries without simplification\cite{cancelliere_1990}, 
as most of the operation is local and thus it is ideal for parallel implementation. Due to the statistical physics background, it is
easy to simulate multi-physics processes including multi-phase and multi-component flow,
evaporation, condensation as well as cavitation. \cite{LBMODELING}. These advantages make the LB method an
ideal numerical tool to study the flow properties in porous media. In this study, we use a multiple-relaxation-time (MRT) based multi-component lattice Boltzmann method with an optimised extension for multi-component Stokes flow in porous media\cite{tolke_2006}. This new approach can simulate binary fluid system with higher viscosity ratios and lower capillary number with higher numerical stability \cite{ahrenholz_2008}. 

For the boundary conditions, we use the simple half-way bounce back scheme which offer second order accuracy and conserve the mass of all components. To simulate the spontaneous imbibitions, a modification on the geometry was performed. We take the micro-model with a squared pore in the centre as an example to show the initial configuration of the simulation. This configuration was showed in Figure (\ref{LB_ini}), the green represent the solid, the colour blue and red represent air and decane respectively. A big reservoir which connect the top three channels was built for both decane and air at the bottom, periodic boundary conditions were applied in X,Y and Z direction.   
\begin{figure}[H]
\begin{center}
        \includegraphics[width=2in]{LB_initial.eps}
\end{center}
\caption{The initial configuration for multi-component simulation}\label{LB_ini}
\end{figure}

For the main displacement process of two-component flow in micro-models, the Reynolds number is very small, the inertia effect is neglectable. The density of decane and air in the simulation was set as equal.  Another important dimensionless parameter is the capillary number which represents the relative effect of viscous force versus surface tension on the interfaces:

\begin{equation}
        Ca=\frac{u_w \mu_w}{\sigma}
        \end{equation}

where $u_w$ and $\mu_w$ are the Darcy velocity and dynamic viscosities of the wetting phase. $\sigma$ is the value of surface tension.  We use a low viscosity ratio of 10 in the simulation rather than the the real decane/air system viscosity ratio of 50. The typical capillary number for the flow in micro-models is of order of $10^-5$ which means the capillary force dominate the displacement process. Therefore the ratio of 10 or ever lower is sufficient to reproduce the main physical process \cite{ahrenholz_2008}. The contact angle is set to $30^\circ$. The implementation of contact angle can be found in \cite{tolke_2002}. The simulation kernel was parallelized  using a distributed memory approach with the Message Passing Interface(MPI).  

The initial distribution of decan and air was shown in Figure (\ref{LB_ini}). Because the micro-model is oil wetting, the decane will be spontaneously imbibed into the model due to the capillary pressure. A video of the process can be captured by the high speed camera for the comparison with the LB simulations.  

\section{Simulation results and comparisons with experimental data}
In both micro-models, the wetting phase (decane) has spontaneously imbibed into the model from top right corner (the single junction micro-model) and bottom right corner (the Berea micro-model), according to the capillary entry pressure rules used by network modelling, the fluid should enter the smallest channel which has the highest capillary pressure first, however, in our study, this is not the case. The results for the single junction micro-model and the Berea micro-model obtained by experiments and the parallel  lattice Boltzmann simulation are shown in Figure (\ref{res1}) and Figure (\ref{res2}) respectively. The snapshots are taken from the top of the micro-models (Z direction). The black-white snapshots are experimental data, the light grey and dark grey represent decan and air respectively, the interface was shown in dark black. The lattice Boltzmann simulations were shown with colours, the colour red, blue and green represent decane, air and PMMA base respectively. 

\begin{figure}[H]
\begin{center}
        \includegraphics[width=3in]{res2a.eps}
\end{center}
\caption{Snapshots in various moments of spontaneous imbibition of decane in a single junction micro-model, experiments results and the lattice Boltzmann simulations. The experimental data and simulation results are shown together to easier the comparison. The left black-white snapshots are experimental data, the colour snapshots are obtained from the LB simulation.}\label{res1}
\end{figure}

\begin{figure}[H]
\begin{center}
        \includegraphics[width=3in]{res3a.eps}
\end{center}
\caption{Snapshots in various moments of spontaneous imbibition of decane in a Berea sandstone micro-model}\label{res2}
\end{figure}

In Figure (\ref{res1}), as can be seen, the decane was imbibed into bottom right corner first, then top left and bottom left corner which is not consistent with the filling rule used in network modelling(Figure\ref{res1}.3, \ref{res1}.4). In the network modelling, the fluid should be imbibed into all the other channels simultaneously. We think this inconsistent was caused by the unsymmetry of the micro-model. As a result, we slightly revise the geometry for the LB simulation to break the symmetry. The width of the bottom right channel is one lattice smaller than all the other channels, and the bottom left corner of the squared pore was on lattice smaller than the other corners. The results of the LB simulation on this modified geometry and the experiments show a very good agreement(Figure \ref{res1}).  Almost all the main process and interface movements are captured by the LB simulation. It should be noted that after the modification on the geometry, the width of the top left and bottom left channel are the same, according to the filling rules of network modelling the decane should enter two channels simultaneously; however, both the experiment and simulation showed that the decane imbibes into the top left channel first as a result of interface contact with the top left corner in advance (Figure \ref{res1}.5,\ref{res1}.6).   

For the imbibition in Berea sandstone micro-model (Figure \ref{res2}), the results from the experiment  and the LB simulation show a very good agreement; most of the main process in the experiment have been achieved by the LB simulation. It can be observed that the decane imbibed from the bottom right corner entered the nearest top right channel first, after that it entered the top left channel which has the highest capillary pressure. This result supported again our hypothesis, for the case of imbibition, the local geometry of the network model junction rather than the capillary pressure of the channels, may play an important role. 

Although the agreement of displacement of interfaces between the simulations and the experiments are quite satisfying, the time scales did not match well. This should be due to the neglect of density ratio between decan and air. The approximation of density ratio equals 1 is based on the assumption of low Reynolds number, however, the Reynolds number increases dramatically when the interface of decane touch the solid wall, as a result, the inertia plays an important role at that moment, the density ratio is no long neglectable. 


\section{Conclusions and outlook}
We investigate the spontaneous imbibitions of decane/air system in two micro-models with increasing complexity. Experiments and the LB simulations were performed to study the flow in these micro-models. The LB simulations successfully simulate 3D displacement of decane/air system, the observed displacement process were well matched. The results demonstrate that the capillary filling rules used in the network modelling may not valid for the case of imbibtion, instead the local geometry play a major role. The comparison illustrate that the LB simulations is promising and quite adequate for multi-component fluid flow simulation in porous media.


We would like to acknowledge that this study is financed by Qatar Carbonates and Carbon Storage Research Centre (QCCSRC). The QCCSRC is funded jointly by Qatar Petroleum, Shell and the Qatar Science \& Technology Park. We thank the High Performance Computing Service of Imperial College London for providing the computing time and technical support.

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